A Generalised Dynamical System, Infinite Time

We identify a number of theories of strength that of Π 1 CA0. In particular: (a) the theory that the set of points attracted to the origin in a generalised transfinite dynamical system of any n-dimensional integer torus exists; (b) the theory asserting that for any Z ⊆ ω and n, the halting set H n of infinite time n-register machine with oracle Z exists. Suppose f : N −→ N. We are going to consider transfinite iterations of such f : N −→ N as a generalised dynamical system. If one wishes, one may think of f acting on the points of an n-dimensional lattice torus where we identify∞ with 0. We set this up as follows. Given a point r = (r1, . . . , rn) ∈ N set: r = (r 1, . . . , r 0 n) = (r1, . . . , rn); r = (r 1 , . . . , r α+1 n ) = f((r α 1 , . . . , r α n)); r = (r 1 , . . . , r λ n) = (Liminf ∗ α→λ r α 1 ,Liminf ∗ α→λ r α 2 , . . . ,Liminf ∗ α→λ r α n) where we define Liminfα→λ r α 1 = Liminfα→λ r α 1 if the latter is < ω, and set it to 0 otherwise, thus: r i = Liminfα→λ r α i if the latter is < ω = 0 otherwise. We may ask after the behaviour of points under this dynamic. For example which points ultimately end up at the origin O? As a more amusing example let p = (p0, p1, p2) ∈ (N) be a triple of three points on the n-dimensional lattice. In general they thus form a proper triangle. Then define: Tf = {(p0, p1, p2) ∈ N | ∃α p0 = p1 = p2 }. Tf is thus the set of possible starting triangles, which at some point collapse and become coincident after iteration of their vertices (and remain collapsed of course from some point α0 onwards). ?